Principal homogeneous space

A principal $G$- object in the category of algebraic varieties or schemes. If $S$ is a scheme and $\mathrm{\Gamma }$ is a group scheme over $S$, then a principal $G$- object in the category of schemes over $\mathrm{\Gamma }$ is said to be a principal homogeneous space. If $S$ is the spectrum of a field $k$( cf. Spectrum of a ring) and $\mathrm{\Gamma }$ is an algebraic $k$- group (cf. Algebraic group), then a principal homogeneous space over $\mathrm{\Gamma }$ is an algebraic $k$- variety $V$ acted upon (from the left) by $\mathrm{\Gamma }$ such that if $k$ is replaced by its separable algebraic closure $\bar{k}$, then each point $v\in V(\bar{k})$ defines an isomorphic mapping $g\to gv$ of the varieties $V_{\bar{k}}$ and ${\mathrm{\Gamma }}_{\bar{k}}$. A principal homogeneous space $V$ is trivial if and only if $V(k)$ is non-empty. The set of classes of isomorphic principal homogeneous spaces over a smooth algebraic group $\mathrm{\Gamma }$ can be identified with the set of Galois cohomology $H^1(k,\mathrm{\Gamma })$. In the general case the set of classes of principal homogeneous spaces over an $S$- group scheme $\mathrm{\Gamma }$ coincides with the set of one-dimensional non-Abelian cohomology $H^1(S_T,\mathrm{\Gamma })$. Here $S_T$ is some Grothendieck topology on the scheme $S$[2].

Principal homogeneous spaces have been computed in a number of cases. If $k$ is a finite field, then each principal homogeneous space over a connected algebraic $k$- group is trivial (Lang's theorem). This theorem also holds if $k$ is a $p$- adic number field and $\mathrm{\Gamma }$ is a simply-connected semi-simple group (Kneser's theorem). If $\mathrm{\Gamma }={\mathrm{\Gamma }}_{m,S}$ is a multiplicative $S$- group scheme, then the set of classes of principal homogeneous spaces over $\mathrm{\Gamma }$ becomes identical with the Picard group $\mathrm{Pic}(S)$ of $S$. In particular, if $S$ is the spectrum of a field, this group is trivial. If $\mathrm{\Gamma }={\mathrm{\Gamma }}_{a,S}$ is an additive $S$- group scheme, then the set of classes of principal homogeneous spaces over $\mathrm{\Gamma }$ becomes identical with the one-dimensional cohomology group $H^1(S,{\mathcal{O}}_S)$ of the structure sheaf ${\mathcal{O}}_S$ of $S$. In particular, this set is trivial if $S$ is an affine scheme. If $k$ is a global field (i.e. an algebraic number field or a field of algebraic functions in one variable), then the study of the set of classes of principal homogeneous spaces over an algebraic $k$- group $\mathrm{\Gamma }$ $$ is based on the study of the Tate–Shafarevich set $⨿⨿(\mathrm{\Gamma })$, which consists of the principal homogeneous spaces over $\mathrm{\Gamma }$ with rational points in all completions $k_V$ with respect to the valuations of $k$. If $\mathrm{\Gamma }$ is an Abelian group over the field $k$, then the set of classes of principal homogeneous spaces over $\mathrm{\Gamma }$ forms a group (cf. Weil–Châtelet group).

References[edit]

Comments[edit]

The notion of a principal homogeneous space is not restricted to algebraic geometry. For instance, it is defined in the category of $G$- sets, where $G$ is a group. Let $G$ be a finite (profinite, etc.) group. Let $E$ be a $G$- set, i.e. a set $E$ with an action $G\times E\to E$ of $G$ on it. Let $\mathrm{\Gamma }$ be a $G$- group, i.e. a group object in the category of $G$- sets, which means that $\mathrm{\Gamma }$ is a group and that the action of $G$ on $\mathrm{\Gamma }$ is by group automorphisms of $\mathrm{\Gamma }$: $(xy{)}^{\gamma }=x^{\gamma }{y}^{\gamma }$ for $\gamma \in G$, $x,y\in \mathrm{\Gamma }$. One says that $\mathrm{\Gamma }$ operates compatibly with the $G$- action from the left on $E$ if there is a $\mathrm{\Gamma }$- action $\mathrm{\Gamma }\times E\to E$ on $E$ such that $(\gamma x{)}^g=({\gamma }^g)(x^g)$ for $g\in G$, $\gamma \in \mathrm{\Gamma }$, $x\in E$. A principal homogeneous space over $\mathrm{\Gamma }$ in this setting is a $G$- set $P$ on which $\mathrm{\Gamma }$ acts compatibly with the $G$- action and such that for all $x,y\in P$ there is a $\gamma \in \mathrm{\Gamma }$ such that $y=\gamma x$. (This is the property to which the word "principal" refers; one also says that $P$ is an affine space over $\mathrm{\Gamma }$.) In this case there is a natural bijective correspondence between $H^1(G,\mathrm{\Gamma })$ and isomorphism classes of principal homogeneous spaces over $\mathrm{\Gamma }$ and, in fact, $H^1(G,\mathrm{\Gamma })$( for non-Abelian $\mathrm{\Gamma }$) is sometimes defined this way.